Relations between $e$ and $π$: Nilakantha's series and Stirling's formula

TL;DR

The paper investigates near-coincidences between the fundamental constants $e$ and $π$ by using classical series and asymptotic expansions. It shows how transforming Nilakantha's series for $π$ to accelerated forms and comparing with the inverse-factorial expansion of $e$ explains the approximate relation $e+2π \\approx 9$, and it derives additional near-equalities such as $π^4+π^5 \\approx e^6$ and $π^9/e^8 \\approx 10$. Employing Stirling's series, the work extracts further approximate identities (e.g., $π^2+π \\approx 13$, Archimedean-type bounds $π \\approx 22/7$ and $π \\approx \\sqrt{51}-4$) and demonstrates how these expansions yield consistent near-equalities like $e^8 \\approx 96π^3$ and relate to $e^π$ and theta-function sums. Overall, the article provides a structured, method-based exploration of $e$–$π$ relations, illustrating how classical tools can uncover and rationalize near-coincidences with potential cross-links to number theory and mathematical constants.

Abstract

Approximate relations between $e$ and $π$ are reviewed, some new connections being established. Nilakantha's series expansion for $π$ is transformed to accelerate its convergence. Its comparison with the standard inverse-factorial expansion for $e$ is performed to demonstrate similarity in several first terms. This comparison clarifies the origin of the approximate coincidence $e+2π\approx 9$. Using Stirling's series enables us to illustrate the relations $π^4+π^5 \approx e^6$ and $π^{9}/e^{8} \approx 10$.The role of Archimede's approximation $π=22/7$ is discussed.